28 XII TOPOLOGY AND TOPOLOGICAL ALGEBRA (12.7.11) For every sequence (jcn) in R, lim inf xn (resp. lim sup xn) is the least H-»00 W~*00 (resp. the greatest) of the cluster values of the sequence (xn). Let a = lim inf xn , and suppose first that a is finite. If c < a < c' and if m is n-+oo any integer, there exists n > m such that c < inf xn+p < c', hence there exists p^O p ^ 0 such that c < xn+p < c'. This shows that a is a cluster value of the sequence (xn) (3.13.11). If fl~ +00 (resp. a= — oo), the argument is the same if we replace c' (resp. c) by a. Conversely, if b is a cluster value of (xn)9 then it is also a cluster value of every sequence (xn+p)p^Q9 and therefore b £ infxn+p (3.13.7). Hence b ^ a. p£0 This result shows that the existence of a limit- of the sequence (xn) is equivalent to the relation lim inf xn = lim sup xn 9 and that the common value tt-»00 7I-+00 of the two sides is then the limit of (xn) (3.16.4). It follows from the definition that if (xni) is any subsequence of a sequence (xn) in R, then we have (1 2.7.1 2) lim inf xnk ^ lim inf xn . fc-»oo -n-»oo (12.7.13) Let E be a Hausdorff topological space and let f be a mapping ofE into R which is lower semicontinuous at a point aeE. Then for every sequence of points ofE such that lim xn == a, we have (12.7.13.1) lim inf /(*„) ^ /(a). H-+00 For every a <f(d) there exists a neighborhood V of a in E such that f(y) ^ a for all y e V; but there exists n0 such that xn e V for all n^n0. Hence f(xn) ^ a for all n^n0) which proves the result. PROBLEMS 1. Let E be a topological space. For each closed subset A of the product space E x ft, show that the mapping #h->inf (A(x)) ;of priA into ft is lower semicontinuous. Conversely, if/: E ~>ft is lower semicontinuous, then the subset B of E x ft consisting of all pairs (x, y) such that f(x) <£ y is closed in E x ft. M-*OO \JpisO / W-+00 n-*cxD Vp^O /